Nowhere commutative semigroup

In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab = ba then a = b.[1] A semigroup S is nowhere commutative if and only if any two elements of S are inverses of each other.[1]

Characterization of nowhere commutative semigroups

Nowhere commutative semigroups can be characterized in several different ways. If S is a semigroup then the following statements are equivalent:[2]

  • S is nowhere commutative.
  • S is a rectangular band (in the sense in which the term is used by John Howie[3]).
  • For all a and b in S, aba = a.
  • For all a, b and c in S, a2 = a and abc = ac.

Even though, by definition, the rectangular bands are concrete semigroups, they have the defect that their definition is formulated not in terms of the basic binary operation in the semigroup. The approach via the definition of nowhere commutative semigroups rectifies this defect.[2]

To see that a nowhere commutative semigroup is a rectangular band, let S be a nowhere commutative semigroup. Using the defining properties of a nowhere commutative semigroup, one can see that for every a in S the intersection of the Green classes Ra and La contains the unique element a. Let S / L be the family of L-classes in S and S / R be the family of R-classes in S. The mapping

ψ : S → (S / R) × (S / L)

defined by

aψ = (Ra , La)

is a bijection. If the Cartesian product (S / R) × (S / L) is made into a semigroup by furnishing it with the rectangular band multiplication, the map ψ becomes an isomorphism. So S is isomorphic to a rectangular band.

Other claims of equivalences follow directly from the relevant definitions.

gollark: If nobody has come up with any philosophy before you, you'll get recognized for saying basically anything.
gollark: The problem is low-hanging fruit.
gollark: At the very least, bronze age governments probably weren't able to do very much of that stuff.
gollark: What? No.
gollark: You would need to be in a vacuum and not subject to significant gravity, but whatever.

See also

Special classes of semigroups

References
  1. A. H. Clifford, G. B. Preston (1964). The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society (p.26). ISBN 978-0-8218-0272-4
  2. J. M. Howie (1976). An Introduction to Semigroup Theory. LMS monographs. 7. Academic Press. p. 96.
  3. J. M. Howie (1976). An Introduction to Semigroup Theory. LMS monographs. 7. Academic Press. p. 3.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.